Jaynes explains that Gibbs entropy is a conserved quantity, for the same reason as the Louiville theorem that conserves the hyper-volume in phase space of a cloud of particles as it traverses its trajectory.

Boltzmann entropy increases. We can show that this is a consequence of quantal interactions during particle collisions, which deny the claim of microscopic irreversibility and erase the path information in the gas particles that would be needed to support Loschmidt's objection to the Boltzmann H-Theorem

In the writer's 1962 Brandeis lectures on statistical mechanics, the Gibbs and Boltzmann
expressions for entropy were compared briefly,
and it was stated that the Gibbs formula gives
the correct entropy, as defined in phenomenological
thermodynamics, while the Boltzmann H
expression is correct only in the case of an ideal
gas. However, there is a school of thought which
holds that the Boltzmann expression is directly
related to the entropy, and the Gibbs' one simply
erroneous. This belief can be traced back to the
famous Ehrenfest review article, which severely
criticized Gibbs' methods.

While it takes very little thought to see that
objections to the Gibbs II are immediately refuted
by the fact that the Gibbs canonical ensemble
does yield correct thermodynamic predictions,
discussion with a number of physicists
has disclosed a more subtle, but more widespread,
misconception. The basic inequality of the
Gibbs and Boltzmann H functions, to be derived
in Sec. II, was accepted as mathematically correct; but it was thought that, in consequence of
the "laws of large numbers" the difference between
them would be practically negligible in
the limit of large systems.

Now it is true that there are many different
entropy expressions that go into substantially
the same thing in this limit; several examples
were given by Gibbs. However, the Boltzmann
expression is not one of them; as we prove in
Sec. Ill , the difference is a direct measure of the
effect of interparticle forces on the potential
energy and pressure, and increases proportionally
to the size of the system.

Failure to recognize the fundamental role of
the Gibbs H function is closely related to a much
deeper confusion about entropy, probability,
and irreversibility in general.

Gibbs' entropy is a constant because the loss of macroscopic order is conserved in the path information of the particles

For example, the
Boltzmann H theorem is almost universally
equated to a demonstration of the second law of
thermodynamics for dilute gases, while ever
since the Ehrenfest criticisms, it has been
claimed repeatedly that the Gibbs H cannot be
related to the entropy because it is constant in
time.

Closer inspection reveals that the situation is
very different. Merely to exhibit a mathematical
quantity which tends to increase is not relevant
to the second law unless one demonstrates that
this quantity is related to the entropy as measured
experimentally. But neither the Gibbs nor
the Boltzmann H is so related for any distribution
other than the equilibrium (i.e., canonical)
one. Consequently, although Boltzmann's H
theorem does show the tendency of a gas to go
into a Maxwellian velocity distribution, this is
not the same thing as the second law, which is a
statement of experimental fact about the direction
in which the observed macroscopic quantities
(P,V,T) change.

The idea of classical coarse-graining takes on new significance with the minimal phase-space volumes of the quantum mechanical uncertainty principle.

Past attempts to demonstrate the second law
for systems other than dilute gases have generally
tried to retain the basic idea of the Boltzmann
H theorem. Since the Gibbs H is dynamically
constant, one has resorted to some kind of coarse-graining
operation, resulting in a new quantity
Ħ, which tends to decrease. Such attempts cannot
achieve their purpose, because (a) mathematically,
the decrease in Ħ is due only to the
artificial coarse-graining operation and it cannot,
therefore have any physical significance; (b) as
in the Boltzmann H theorem, the quantity whose
increase is demonstrated is not the same thing
as the entropy. For the fine-grained and coarse-grained
probability distributions lead to just the
same predictions for the observed macroscopic
quantities, which alone determine the experimental
entropy; the difference between H and Ħ
is characteristic, not of the macroscopic state,
but of the particular way in which we choose to
coarse-grain. Any really satisfactory demonstration
of the second law must therefore be based on
a different approach than coarse-graining.

Actually, a demonstration of the second law,
in the rather specialized situation visualized in
the aforementioned attempts, is much simpler
than any H theorem. Once we accept the well-established
proposition that the Gibbs canonical
ensemble does yield the correct equilibrium
thermodynamics, then there is logically no room
for any assumption about which quantity represents
entropy; it is a question of mathematically
demonstrable fact. But as soon as we have understood
the relation between Gibbs' H and the
experimental entropy, Eq. (17) below, it is
immediately obvious that the constancy of
Gibbs' H, far from creating difficulties, is precisely
the dynamical property we need for the
proof.

It is interesting that, although this field has
long been regarded as one of the most puzzling
and controversial parts of physics, the difficulties
have not been mathematical. Each of the above
assertions is proved below or in the Brandeis
lectures, using only a few lines of elementary
mathematics, all of which was given by Gibbs.
It is the enormous conceptual difficulty of this
field which has retarded progress for so long.
Readers not familiar with recent developments
may, I hope, be pleasantly surprised to see how
clear and basically simple these problems have
now become, in several respects. However, as we
will see, there are still many complications and
unsolved problems.

Inspection of several statistical mechanics
textbooks showed that, while most state the
formal relations correctly, their full implications
are never noted. Indeed, while all textbooks give
extensive discussions of Boltzmann's H, some
recent ones fail to mention even the existence of
the Gibbs H. I was unable to find any explicit
mathematical demonstration of their difference.
It appeared, therefore, that the following note
might be pedagogically useful.

According to Jaynes (and Gibbs), information is conserved when macroscopic order disappears because it simply changes into microscopic (thus invisible) order as the path information of all the gas particles is preserved. As Boltzmann's mentor Joseph Loschmidt had argued in the early 1870's, if the velocities of all the particles could be reversed at an instant, the future evolution of the gas would move in the direction of decreasing entropy. All the original order would reappear.

This is consistent with the idea of Pierre-Simon Laplace's super-intelligent demon and completely deterministic laws of nature. It also follows from the Louville theorem that the hyper-volume of a cloud of points in phase space is a constant as the system evolves. Classical mechanics and physical determinism was shown to be only an approximation for large numbers of particles shortly after Gibbs's death by Albert Einstein and the later "founders" of quantum mechanics.

When quantum effects are included in the collision of gas particles, Boltzmann's idea of "molecular disorder" is seen to be correct and path information is destroyed.

Nevertheless, Gibbs's idea of the conservation of information is still widely held today by mathematical physicists. And most texts on statistical mechanics still claim that microscopic collisions between particles are reversible. Some explicitly claim that quantum mechanics changes nothing, but that is because they limit themselves to the unitary (conservative and deterministic) evolution of the Schrödinger equation and ignore the collapse of the wave function.

For example, Richard Tolman (p.8) claimed that the “principle of dynamical reversibility” holds also in quantum mechanics in appropriate form, indicating that quantum theory supplies no new kind of element for understanding the actual irreversibility in the macroscopic behavior of physical systems.

And D. ter Haar (p. 292) said “The transition from classical to statistical mechanics does not introduce any fundamental changes.”

This is because both classical and quantum statistical mechanics describe ensembles of systems. Such systems are in “mixed states,” disregarding the interference terms in the density matrix of the “pure states” density operator. This is the basis for decoherence theories.

The origin of irreversibility depends on the ontological chance involved in von Neumann's Process 1, Dirac's projection postulate, the "collapse of the wave function," denied by so many interpretations of quantum mechanics and ignored in statistical mechanics texts.

In her 2008 book, Carolyne Van Vliet (p.678) says that the theory of non-equilibrium statistical mechanics is incomplete without some kind of randomization at the microscopic level.

Jaynes likely did not accept the collapse of the quantum mechanical wave function. He was strongly influenced by Eugene Wigner, who was an early denier of the projection postulate and supporter of the unitary evolution of the universal wave function. He says

I have profited from discussions of these problems, over many years, with Professor E. P. Wigner, from whom I first heard the remark, "Entropy is an anthropomorphic concept."